I need to show 1^3 + 2^3 + ... + n^3 < (1/2)n^4 for all n in N and n >=3.

I want to use mathematical induction, but I don't know if I need to use the first Mathematical induction or the second one?

Thanks

February 3rd 2010, 07:34 PM

Archie Meade

Quote:

Originally Posted by inthequestofproofs

I need to show 1^3 + 2^3 + ... + n^3 < (1/2)n^4 for all n in N and n >=3.

I want to use mathematical induction, but I don't know if I need to use the first Mathematical induction or the second one?

Thanks

If

then hopefully

Proof

Then

which is less than

Hence, if

then is certainly <

The inductive process is validated for the hypothesis.

True for n=3 ?

Proven

February 4th 2010, 09:45 AM

novice

We are to prove that , for

The sum of cube series is equal to

We can restate the question as follows:

, for

Proof:

Base case: For integer ,

Induction hypothesis: Suppose for every integer

Then

LHS:

=

=

=

=

RHS:

=

Putting LHS and RHS together:

Hence, by induction hypothesis, , for all in

February 4th 2010, 12:17 PM

inthequestofproofs

question

After you put the LHS and RHS together, how do you simplify the inequality?\

thanks for your detailed work

February 4th 2010, 01:34 PM

novice

Simple algebra. Move the 1st and 2nd terms to the right hand side, and move the last term from the right hand side to the left, etc.

February 4th 2010, 01:39 PM

inthequestofproofs

question about inequality

THanks!! it makes sense. My new question is that if what we are trying to prove the < (inequality), it seems that the RHS < LHS, because some terms are larger than others. However, there are other terms that are smaller than others. So, even though it seems logical that the inequality is true, is it enough to state it like that, or is there a need of more explanation to justify the inequality?

February 4th 2010, 03:09 PM

novice

Quote:

Originally Posted by inthequestofproofs

THanks!! it makes sense. My new question is that if what we are trying to prove the < (inequality), it seems that the RHS < LHS, because some terms are larger than others. However, there are other terms that are smaller than others. So, even though it seems logical that the inequality is true, is it enough to state it like that, or is there a need of more explanation to justify the inequality?

That's all.

If the base case and induction hypothesis are true, then proposition is true for all positive integers---end of proof.